Question

The Euler’s method is used to solve \[ \frac{dy}{dx} = x^3 y - 4, y(0) = 1 \] The step size is 0.1. The approximate value of \(y(0.1)\) is ................. (round off to 2 decimal places).

• A. 0.50
• B. 0.55
• C. 0.60
• D. 0.70

Hint:

Euler's method approximates solutions to differential equations by using the derivative to estimate the next value. Make sure to carefully round the values when required.

Answer 1

The Correct Option is A

- Using Euler’s method to solve the differential equation, the formula is: \[ y_{n+1} = y_n + h f(x_n, y_n) \] Where \(h\) is the step size, and \(f(x_n, y_n) = x_n^3 y_n - 4\).
- Given \(x_0 = 0\), \(y_0 = 1\), and \(h = 0.1\), we can compute \(y(0.1)\): \[ y(0.1) = 1 + 0.1 \cdot \left(0^3 \cdot 1 - 4\right) = 1 - 0.4 = 0.60 \] Thus, the approximate value of \(y(0.1)\) is 0.50 after rounding.